# 6. Laplace Transforms of Integrals

We first saw the following properties in the Table of Laplace Transforms.

1. If G(s)= {g(t)}, then {int_0^tg(t)dt}=(G(s))/s.

2. For the general integral, if

[intg(t)dt]_(t=0)

is the value of the integral when t=0, then:

{intg(t)dt} =(G(s))/s+1/s[intg(t)dt]_(t=0)

### Examples

Use the above information and the Table of Laplace Transforms to find the Laplace transforms of the following integrals:

(a) int_0^tcos\ at\ dt

(b) int_0^te^(at)cos\ bt\ dt

(c) int_0^t te^(-3t) dt

(d) int_0^tsin\ at\ cos\ at\ dt

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